Critical Mach versus Altitude...

[TedUnderwood]

I'm trying to determine what happens to Critical Mach in relation to altitude. I cannot find any sources and I've tried reading thru High Speed Aerodymics in Aerodynamics for Naval Aviators but haven't had much luck. I know in some aircraft VMO/MMO goes up slightly with altitude and then comes down. Is it dependant on the design of the aircraft or can it be plotted as it relates to mach speed?

[FE Hoppy]

Mcrit again!

S.E.A.R.C.H

[varigflier]

There is an explanation in the book "Fly the Wing".
I hope this helps.

VF

[Old Smokey]

Mcrit is the same at ALL Altitudes.:ok:

No further explanation necessary.

Regards,

Old Smokey

[TedUnderwood]

Thanks for the replies, I tried the search function, and there were some good threads, but none that referenced the affects related to altitude.

I guess I didn't ask the question properly. If Mcrit stays the same, and mach speed (in relation to speed over the ground plotted against altitude) changes with altitude, then Mcrit follows the same line as Mach speed versus altitude...correct?

I guess the only follow up I have for Old Smokey is; if VMO/MMO changes with altitude, doesn't it follow that Mcrit would change as well (ie, in relation to ground speed with no wind, just raw speed over the ground)?

PS, I'm glad I dragged Old Smokey into this! When I was searching in google I kept finding his posts in here but most were dated back to 2004/05. Nice to see he's still at it 5 years later! Thanks!

[FE Hoppy]

Mcrit is a Mach number of trivial significance which is related to Mdd which is much more interesting and changes depending on Cl amongst other things. Now if you were to climb at a constant Mach number it's fair to say that as density reduces CL has to increase and therefore both Mdd and Mcrit will reduce but they are not reducing as a function of altitude but rather of Cl.

[enicalyth]

http://www.fzt.haw-hamburg.de/pers/Scholz/paper/DGLR-2005-122Paper.pdf

Sadly this paper does slip up a little inasmuch as the authors have gone for the formulae and skipped the authors' caveats.

Hoppy alludes to other factors which will include wing sweep, airfoil characteristics, thickness/chord ratio. In the paper I allude to please note that the analysis is entirely empirical. Alas that is where these empiricists themselves went a bit wrong. They thought everyone else was using the same empiricism!

Try Stanford Uni for Shevell's program on compressibility, Mdd, Mcrit. I seem to recall his work was based on conventional aircraft but without the tail in a wind tunnel. He made an allowance for tail down force, which no small matter the paper reviewers missed in their haste to grab the formulae. Tisk Tisk.

Lumme its all a long time ago now.

Hope this is helpful and gives you are sources to search for.

The "E"

[profsx]

This is why Mcrit should change with change in altitude->

At FL200 ,Assume Aerofoil 'A' 's airflow reaches M1.0 on the upper surface at forward speed of M 0.85, This defines Mcrit of A as Mcrit=M0.85, which means the upper surface of the wing has accelerated the air by M0.15.

At FL400, the air is thinner, ie, less dense, ie, is easier to accelerate (F=mXa, where m has dropped due to drop in density). Because the air is easier to accelerate, it would accelerate over the upper surface of the wing quicker, which means the upper surface can accelerate the airflow by, say, M0.16, effectively dropping Mcrit to M0.84 (For arguments sake)

Conclusion-Mcrit should ideally drop, but it probably drops insignificantly, so it is probably calculated at the service/absolute ceiling and is fed to us pilots as one number easier to remember.....

My language should have made it clear that it is a theory backed by nothing but previously studied material.

[Pugilistic Animus]

the precise IMN for Mcrit really all depends on the little twists and turns taken by the designers:)

[PBL]

It would be nice to see an answer to the question originally posed. So here is one. I must say I don't find the suggested HAW-Hamburg paper very helpful for answering the question, but then the authors were explaining something they felt was new, not basic aero to be found in textbooks.

I'm trying to determine what happens to Critical Mach in relation to altitude. I cannot find any sources and I've tried reading thru High Speed Aerodymics in Aerodynamics for Naval Aviators but haven't had much luck.

The coefficient of pressure, Cp, is defined as the increment of pressure at that point on the airfoil (p.local-p.freestream) divided by the freestream dynamic pressure. Since it is a function of point x, let us denote it Cp(x). It follows from the compressible Bernoulli equation that Cp(x) is an arithmetic function (that is, a function which is a composition of addition, subtraction, multiplication, division, and exponent) of Mach number at that point on the airfoil, M.local(x), freestream Mach number, M.freestream, and gamma, a constant with value of about 1.4 for air. The only varying quantities for flight are M.local(x) and M.freestream.

Now, there is some point x0 on any airfoil at which Cp at that point, Cp(x0), is a minimum. Call that minimum simply Cp.0, to conform with convention. x is also a point of maximum velocity on the airfoil surface. There may be more than one x0 (indeed, for a constant-form airfoil there is going to be a line of them).

You can find out x0, and Cp.0, by experiment in incompressible conditions if you wish. Indeed, let Cp(y).0 denote Cp(y) in incompressible conditions for any point y on the airfoil.

According to the Prandtl-Glauert approximation, Cp at any point y is going to vary with freestream Mach number as Cp(y).0 / sqrt(1 - (M.freestream)^2). Now, Prandtl-Glauert is experimental (derived by curve-fitting on experimental data), but is pretty good up to M=0.7 or 0.8 or so, so I understand. Above that, there are more complicated formulas, but they are still arithmetic functions of Cp and M.freestream.

So x0 is going to be the first point at which M.local = 1, as M.freestream increases. That is because Cp(x0).0 is lower (or equal to) Cp(y).0 for any other point y, and when you divide them both by sqrt(1-(M.freestream)^2), that "less than or equal to" relation is preserved. That point at which M.local(x0)=1 is going to occur at a specific value of M.freestream. That value is known as M.crit.

No mention of altitude, everything is an arithmetic function of M.freestream. That may be why you find no mention of altitude in calculations of M.crit.

I can see why Hurt didn't help much, looking at the couple of sentences on p215 in which he talks about M.crit. Try Introduction to Flight, John D. Anderson Jr., Section 5.9, Sixth Edition, McGraw-Hill 2008. It is a standard reference, and Anderson devotes 12 pp to it, which means you get a lot more in the way of careful derivation and explanation.

I make no claim that this explanation is particularly intuitive, because that depends on one's favorite kind of intuition. Please feel free to find a more intuitive explanation than this one.

PBL

[BOAC]

It would be nice to see an answer to the question originally posed. - did you miss Smoker's concise one-liner?:)

[PBL]

Quote:
It would be nice to see an answer to the question originally posed.(End quote)

- did you miss Smoker's concise one-liner?

It was a two liner, neither line of which is correct, in my opinion.

The first line misses the crucial word "approximately", I think. The second is mistaken - one needs to explain how and why it is approximate. My explanation indicated that.

PBL

[TedUnderwood]

Now we're talking! Holy smokes, I can't read this with a beer can in my hand. Thanks for the information, I'll check out the Anderson reference.

[BOAC]

It was a two liner, neither line of which is correct, in my opinion. - actually in Pedantic Bielefeld Land it is a four-liner, two of which are certainly correct (within empirical bounds).

[Pugilistic Animus]

I guess one does not undertand IMN v. TMN...:rolleyes:

And it's the Karman-Tsien relation most frequently used ---more than the Prantdl...:rolleyes:


:zzz:

[HazelNuts39]

PBL;
Your theory is fine for a given "geometry", which term includes a given angle of attack. For an airplane at a given weight, AoA (1g) at constant Mach (free stream) increases with altitude, and hence Cp (absolute).

BTW, according to Wikipedia, Prandtl-Glauert "is derived from linearized equations".

regards,
HN39

[PBL]

Your theory is fine for a given "geometry", which term includes a given angle of attack. For an airplane at a given weight, AoA (1g) at constant Mach (free stream) increases with altitude, and hence Cp (absolute).


It's not really my theory, I am just recounting the thoughts of others.

What do you think of the following? The need to maintain a constant weight in equilibrium (vertically-unaccelerated flight) means constant lift (of magnitude = weight). And this total lift derives from the coefficient of lift c_L (along the chord line). And this c_L derives from integrating the difference between C_p on the upper surface and C_p on the lower surface along the chord line, and then dividing by the chord. So c_L is a function of C_p. So, so is the total lift. All at a given AoA, as you observed. Different AoA, different c_L, different lift; you're accelerating up or down now.

This doesn't seem to fit with what you are suggesting.


BTW, according to Wikipedia, Prandtl-Glauert "is derived from linearized equations".


I find Wikipedia a pretty poor source for understanding almost anything mathematical. To me, a "linearized equation" would mean an equation with only linear terms of its variables. And that is misleading.

From what I understand, Prandtl used the approximation in his lectures in Göttingen as early as 1922. Glauert "derived" the compressibility correction in 1928, and it was used until 1939 when there were better. (Ref: Anderson, Fundamentals of Aerodynamics. It may well be in his history, also, but I don't have that.)

Glauert's derivation starts from a linear partial differential equation (of second order, called the "linearised velocity potential equation), which is an approximation of the velocity potential equation, which is not a linear PDE. He does it by formally transforming the independent variables so that the linear PDE transforms into Laplace's equation (the sum of the second partials on each independent variable, aka the Laplacian, = 0), which is easy to solve; and then one transforms back. But remember that this derivation starts from an approximation in the first place (a linear PDE that "corresponds" to the velocity potential PDE).

The Prandtl-Glauert formula is a "reasonable" approximation for thin airfoils, low AoA and M.freestream < 0.7 or thereabouts.

How do we know all this? Curve-fitting. You have the data. You try a simple quadratic fit, and it does pretty well. Then along comes someone else, who starts from another well-known approximation (which is also necessary for mathematicians to derive anything useful from the original equation), and derives yours from it. Now you can all be even more convinced that it does a good job. Within the region in which it does a good job, of course. Outside that region, it doesn't do a good job (this is a tautology, obviously). That is what curve-fitting is all about. But if someone else wants to use a different phrase for this process, that's OK with me.

PBL

[HazelNuts39]

PBL;
Thank you for your elaboration on the history of the Prandtl-Glauert transformation which I found very interesting while largely confirming what I would have imagined. I discovered Wikipedia only recently, and was surprised by the extent of coverage and mostly excellent quality. By the nature of it, needs to be used with discretion.

Different AoA, different c_L, different lift; you're accelerating up or down now.
Not really. For a given airplane configuration, weight, Mach, and 1g, c_L is inversely proportional to ambient pressure, so it increases with altitude. With increasing AoA the shape of the pressure distribution over the chord changes, in many cases it becomes more 'peaky'.

regards,
HN39

[PBL]

For a given airplane configuration, weight, Mach, and 1g, c_L is inversely proportional to ambient pressure, so it increases with altitude. With increasing AoA the shape of the pressure distribution over the chord changes, in many cases it becomes more 'peaky'.


Give the relationship you have "simplified", and we can discuss it. This formulation is too vague for me to see what you are trying to get at.

For example, I can't see how c_L is dependent on weight. For c_L has a two-dimensional definition, and it is the same whether your wing is one meter span or one hundred, whereas weight is for the entire three-dimensional airplane.

PBL

[HazelNuts39]

Give the relationship you have "simplified", and we can discuss it.c_L=1.3826*10^-4*W/(delta*M^2*S)
where W=weight (mass) in kg, delta=ratio of ambient to SLS pressure, M=Mach, S= (reference) wing area in m^2

For c_L has a two-dimensional definition, and it is the same whether your wing is one meter span or one hundred, whereas weight is for the entire three-dimensional airplane.c_L for the airplane (wing, fuselage, tailplane) is commonly defined as LIFT/(q*S), where q=dynamic pressure=0.5*rho*TAS^2
The formula given above derives from that definition, by replacing TAS by Mach, and inserting standard values at SL for air density, pressure, speed of sound, etc.

regards,
HN39

[PBL]

I am having a little difficulty understanding what you are proposing. Here is why.

c_L=1.3826*10^-4*W/(delta*M^2*S)
where W=weight (mass) in kg, delta=ratio of ambient to SLS pressure, M=Mach, S= (reference) wing area in m^2


Looking at the units, this is ((number) x Force) / (number x number x Area), i.e force per unit area. Let M be mass units, L be length units, T be time units. Force is then M x L / T^2; Area is L^2, so this comes down to M x L / T^2 x L^2, which is M / T^2 x L.

Whereas in the usual definition of c_L, which you quote as c_L ..... is commonly defined as LIFT/(q*S)
c_L has no units: (M x L / T^2) / ( (M / L^3) x (L/T)^2 x L^2 ) all cancels out.

PBL

[HazelNuts39]

PBL;

Yes, you're quite right. c_L is dimensionless. The formula interprets W as mass.

regards,
HN39

PM
The constant just after the equal sign is equal to 2*g/(rho*c^2), where:
g = acceleration of gravity at SL = 9.80665 m/s^2
rho = air density at SL std temp = 1.225 kg/m^3
c = speed of sound at SL std temp = 340.3 m/s

[PBL]

The formula interprets W as mass, all units are accounted for in the number that precedes it, which has the dimension (in your notation) of L^2/M.

Sorry, numbers have no dimensions to me.

Terminology difference. Yes, it is now clear to me that your numbers are constants. However, I still have difficulty with equating weight with mass. Weight is to me still a force (opposed in the case of level flight to lift, and one which I feel clearly every time I pick up my suitcase), so still M x L / T^2.

PBL

[HazelNuts39]

PBL: Sorry I missed your post while editing mine. I trust the re-edit clears things up.

[Rivet gun]

An alternative (and perhaps simpler?) way into this one is to note that

q = 1/2*gamma*M^2*P

where P is staitc pressure, Hence:

c_L = weight/(1/2*gamma*M^2*P*S)

If you substitute P= delta*101325 Pa, weight = mass*9.80665 m/s^2 and gamma=1.4 you get HN39's formula.

In any event c_L (and hence alpha) increase as you cimb at constant Mach. Would Mcrit for a wing increase with alpha? I don't know but I would think it might.

[HazelNuts39]

This graph: Variation of AoA with altitude (https://docs.google.com/leaf?id=0B0CqniOzW0rjMmUyZTQ3ZmQtMzFiOS00YmYwLWI0ZTUtYTI0ZmN hZWYzNGE3&hl=en_GB&authkey=COmHkuAE) shows angle of attack at constant weight and Mach (0.65) for a typical new technology widebody airplane.

regards,
HN39

[Pugilistic Animus]

The Prandtl relation and the more commonly used Karman Tsien have two main applications...
1. determine the lift slope curve dCl/dalpha
2. to find the highest incompressible Lift coefficient where the free stream
MN =1.00,...or Mcrit---usually in a graphical tabulation of some kind

:)